Bonsoir,
1) ensemble de derivabilite : R
f(x) = u * v
u = x^3
u' = 3x^2
v = (x + 1)^2
v' = 2(x + 1)
f '(x) = u'v + uv'
f '(x) = 3x^2(x + 1)^2 + x^3[2(x + 1)]
f '(x) = 3x^2(x + 1)^2 + 2x^3(x + 1)
f '(x) = x^2[3(x + 1)^2 + 2x(x + 1)]
f '(x) = x^2 (x + 1)[3(x + 1) + 2x]
f '(x) = x^2 (x + 1)(3x + 3 + 2x)
f '(x) = x^2 (x + 1)(5x + 3)
2) signe :
x^2 = 0
x = 0
x + 1 = 0
x = -1
5x + 3 = 0
5x = -3
x = -3/5 = -0,6
X....|..-inf........(-1)........(-3/5).....0.......+inf
-------------------------------------------------------
f'(x)|.........(+)....O.....(-)..O....(+).O...(+).......
3) f est croissante sur ]-inf; -1[ U ]-3/5 ; + inf[
f est décroissante sur ] -1 ; -3/5 [
Calcul des maximums/minimums :
f(-1) =
f(-3/5) =
f(0) =
Je te laisse calculer
4) tableau de signe :
x^3 = 0
x = 0
(x + 1)^2 = 0
x + 1 = 0
x = (-1)
X ..........|..-inf...........(-1)..............0.........+Inf
x^3.......|...........(-)........................O....(+)........
(x+1)^2.|...........(+).....O......(+)..........(+)...........
f(x)........|...........(-)......O......(+)....O.....(+)........
f(x) est négative sur ]-inf;-1[
f(x) est positive sur ]-1;+inf[
